Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit combinatorial description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise vertex disjoint e...

The bipartite density of a graphG is max{|E(H)|/|E(G)| : H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph. A biased maximum bipartite subgraph of a graph G is a bipartite subgraph of G with the maximum number of edges such that one of its partite sets is independent in G. Let H denote the collection of all connected cubic graphs ...

In this paper, we study the parameterized complexity and inapproximability of the Induced Matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian cycle in a hamiltonian bipartite graph, the problem is W[1]-hard and cannot be solved in time n 1 2 ) unless W[1]=FPT, where n is the number of vertices in the graph. In addition, we show that unless NP=P, the maximum induc...

In 2010, Vukičević introduced an new graph invariant, the inverse sum indeg index of a graph, which has been studied due to its wide range applications. Let Bnd be class bipartite graphs order n and diameter d. this paper, we mainly characterize in with maximal index. Bipartite largest, second-largest, smallest indexes are also completely characterized.

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given gr...

Let G be a connected bipartite graph. An involution α of G that preserves the bipartition of G is called bipartite. Let Gα be the graph obtained from G by adding to G the natural perfect matching induced by α. We show that the k-cube Qk is isomorphic to the direct product G×H if and only if G is isomorphic to Qk−1 for some bipartite involution α of Qk−1 and H = K2.

A bipartite graph G is called (α, β)-biregular if all vertices in one part of G have the degree α and all vertices in the other part have the degree β. An edge coloring of a graph G with colors 1, 2, 3, . . . , t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored ...

The Regularity Lemma [16] is a powerful tool in Graph Theory and its applications. It basically says that every graph can be well approximated by the union of a constant number of random-looking bipartite graphs called regular pairs (see the definitions below). These bipartite graphs share many local properties with random bipartite graphs, e.g. most degrees are about the same, most pairs of ve...

A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V) is called its signed degree set. In this paper, we prove that every set of ...

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bipartite graph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipa...